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Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications

Michael Ruzhansky (UGent) , Serikbol Shaimardan (UGent) and Kanat Tulenov (UGent)
(2026) JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS. 32(1).
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Abstract
In this paper, we study Hörmander type Fourier multiplier theorem and the Nikolskii inequality on quantum tori. On the way to obtain these results, we also prove some classical inequalities such as the Paley, Hausdorff-Young-Paley, Hardy-Littlewood, and Logarithmic Sobolev inequalities on quantum tori. As applications we establish embedding theorems between Sobolev, Besov spaces as well as embeddings between Besov and Wiener and Beurling spaces on quantum tori. We also analyse B-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an application of the study, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.
Keywords
Quantum tori, H & ouml, rmander Fourier multiplier, Nikolskii inequality, Hausdorff-Young inequality, Logarithmic Sobolev inequality, Besov space, Wiener and Beurling spaces, HARDY-LITTLEWOOD, BERNSTEIN-NIKOLSKII, PALEY INEQUALITIES, HARMONIC-ANALYSIS, INTERPOLATION, SPACES

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MLA
Ruzhansky, Michael, et al. “Hörmander Type Fourier Multiplier Theorem and Nikolskii Inequality on Quantum Tori, and Applications.” JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, vol. 32, no. 1, 2026, doi:10.1007/s00041-025-10217-z.
APA
Ruzhansky, M., Shaimardan, S., & Tulenov, K. (2026). Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 32(1). https://doi.org/10.1007/s00041-025-10217-z
Chicago author-date
Ruzhansky, Michael, Serikbol Shaimardan, and Kanat Tulenov. 2026. “Hörmander Type Fourier Multiplier Theorem and Nikolskii Inequality on Quantum Tori, and Applications.” JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 32 (1). https://doi.org/10.1007/s00041-025-10217-z.
Chicago author-date (all authors)
Ruzhansky, Michael, Serikbol Shaimardan, and Kanat Tulenov. 2026. “Hörmander Type Fourier Multiplier Theorem and Nikolskii Inequality on Quantum Tori, and Applications.” JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 32 (1). doi:10.1007/s00041-025-10217-z.
Vancouver
1.
Ruzhansky M, Shaimardan S, Tulenov K. Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS. 2026;32(1).
IEEE
[1]
M. Ruzhansky, S. Shaimardan, and K. Tulenov, “Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications,” JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, vol. 32, no. 1, 2026.
@article{01KM3528HC6GRQ6CYS0WKN6650,
  abstract     = {{In this paper, we study Hörmander type Fourier multiplier theorem and the Nikolskii inequality on quantum tori. On the way to obtain these results, we also prove some classical inequalities such as the Paley, Hausdorff-Young-Paley, Hardy-Littlewood, and Logarithmic Sobolev inequalities on quantum tori. As applications we establish embedding theorems between Sobolev, Besov spaces as well as embeddings between Besov and Wiener and Beurling spaces on quantum tori. We also analyse B-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an application of the study, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.}},
  articleno    = {{7}},
  author       = {{Ruzhansky, Michael and Shaimardan, Serikbol and Tulenov, Kanat}},
  issn         = {{1069-5869}},
  journal      = {{JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS}},
  keywords     = {{Quantum tori,H & ouml,rmander Fourier multiplier,Nikolskii inequality,Hausdorff-Young inequality,Logarithmic Sobolev inequality,Besov space,Wiener and Beurling spaces,HARDY-LITTLEWOOD,BERNSTEIN-NIKOLSKII,PALEY INEQUALITIES,HARMONIC-ANALYSIS,INTERPOLATION,SPACES}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{49}},
  title        = {{Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications}},
  url          = {{http://doi.org/10.1007/s00041-025-10217-z}},
  volume       = {{32}},
  year         = {{2026}},
}
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